Stochastic Optimal Control in Finance

Preface These are the extended version of the Cattedra Galileiana I gave in April 2003 in Scuola Normale, Pisa. I am grateful to the Society of Amici della Scuola Normale for the funding and to Professors Maurizio Pratelli, Marzia De Donno and Paulo Guasoni for organizing these lectures and their hospitality. In these notes, I give a very quick introduction to stochastic optimal control and the dynamic programming approach to control. This is done through several important examples that arise in mathematical finance and economics. The theory of viscosity solutions of Crandall and Lions is also demonstrated in one example. The choice of problems is driven by my own research and the desire to illustrate the use of dynamic programming and viscosity solutions. In particular, a great emphasis is given to the problem of super-replication as it provides an usual application of these methods. Of course there are a number of other very important examples of optimal control problems arising in mathematical finance, such as passport options, American options. Omission of these examples and different methods in solving them do not reflect in any way on the importance of these problems and techniques. Most of the original work presented here is obtained in collaboration with Professor Nizar Touzi of Paris. I would like to thank him for the fruitful collaboration, his support and friendship.